#72303510

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8512 conjugacy classes of subgroups; only the 3037 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(11,15)(14,16)(17,18)(19,21)(20,23)(22,24)(25,26,28,31)(27,29,32,30);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,27,28,32)(26,29,31,30);(17,19)(18,21)(20,24)(22,23)(25,27)(26,29)(28,32)(30,31);(2,5)(6,7)(10,13)(14,16)(19,24)(21,22)(27,32)(29,30);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^10|a^2=g,b^2=i,c^b=cg,d^b=dj,d^c=dh,e^a=eh,e^b=ei,e^d=ej,f^a=fi,f^e=fj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303534, #72305966, #72306602
maximal normal subgroup of	#72303534, #72305966, #72306602

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);